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Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
114
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Lesson 8
Objective: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
Suggested Lesson Structure
Fluency Practice (12 minutes)
Application Problem (7 minutes)
Concept Development (31 minutes)
Student Debrief (10 minutes)
Total Time (60 minutes)
Fluency Practice (12 minutes)
Sprint: Write Fractions and Decimals 4.NF.5 (9 minutes)
Expanded Form 4.NF.5 (3 minutes)
Sprint: Write Fractions and Decimals (9 minutes)
Materials: (S) Write Fractions and Decimals Sprint
Note: This Sprint reviews Lessons 4–7.
Expanded Form (3 minutes)
Materials: (T/S) Personal white board
Note: This fluency activity reviews Lesson 7.
T: (Write 417
100.) Write 4 and 17 hundredths in expanded fraction form without multiplication.
S: (Write 417
100 = 4 +
1
10 +
7
100.)
T: Write 4 and 17 hundredths in expanded decimal form.
S: (Write 4.17 = 4 + 0.1 + 0.07.)
Repeat the process for 2564
100.
T: (Write 5.93.) Write 5 and 93 hundredths in expanded decimal form.
S: (Write 5.93 = 5 + 0.9 + 0.03.)
Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
115
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T: Write 5 and 93 hundredths in expanded fraction form.
S: (Write 593
100= 5 +
9
10 +
3
100.)
Application Problem (7 minutes)
Jashawn had 5 hundred dollar bills and 6 ten dollar bills in his wallet. Alva had 58 ten dollar bills under her mattress. James had 556 one dollar bills in his piggy bank. They decide to combine their money to buy a computer. Express the total amount of money they have using the following bills:
a. Hundreds, tens, and ones b. Tens and ones c. Ones
Note: This Application Problem reviews expanded form and patterns of ten in the place value chart, as taught in Module 1. Reviewing patterns of ten and decomposition of familiar, larger place value units prepares students for today’s exploration of decomposition and composition of smaller place value units.
Concept Development (31 minutes)
Materials: (T/S) Area model and place value chart (Template), personal white board
Problem 1: Represent numbers in unit form in terms of different units using the area model.
T: (Place the area model and place value chart template into personal white boards.) Show 2 ones 4 tenths shaded on the area model.
T: (Point to the first rectangle.) How many tenths are in 1?
S: 10 tenths.
T: Record 10 tenths below the first two rectangles. (Point to the third rectangle.) How many tenths are represented?
S: 4 tenths.
T: Record 4 tenths below this rectangle. (Write the addition symbol between the units.) What is 10 tenths plus 10 tenths plus 4 tenths?
S: 24 tenths.
Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
116
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T: (Write 2.4.) So, 2 and 4 tenths is equal to 24 tenths, true?
S: True.
T: Shade 2 ones 40 hundredths on the next set of area models.
T: Record an addition sentence in unit form that tells how many hundredths are shaded.
S: (Write 100 hundredths + 100 hundredths + 40 hundredths = 240 hundredths.)
T: What decimal number is 240 hundredths equal to?
S: 2.40. 2.4.
T: How can it be equivalent to both?
S: 4 tenths is equal to 40 hundredths, so 0.4 equals 0.40.
Problem 2: Represent numbers in unit form in terms of different units using place value disks.
T: Represent 2 as tenths. How many tenths are in 2 ones?
S: 1 = 10
10, 2 =
10
10+
10
10=
20
10.
T: Say the equivalence.
S: 2 ones equals 20 tenths.
T: Show 2 ones 4 tenths on your place value chart using place value disks. Express the number in unit form as it is shown on the chart.
S: 2 ones 4 tenths.
T: Decompose the 2 ones, and express them as tenths.
S: 2 ones = 20
10. There are 20 tenths + 4 tenths = 24 tenths.
T: How can I express 24 tenths as hundredths?
S: You can decompose the tenths to hundredths and count the total number of hundredths. That’s too many place value disks to draw!
T: You are right! Let’s solve without drawing place value disks. 1 tenth equals how many hundredths?
S: 1 tenth equals 10 hundredths.
T: 2 tenths is equivalent to how many hundredths?
S: 2 tenths equals 20 hundredths.
T: So, 24 tenths equals…? Discuss it with your partner.
S: 240 hundredths. There are 10 times as many hundredths as there are tenths. We showed that using area models. We can multiply the numerator and denominator by the same number, just like with fractions.
MP.6
Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
117
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NOTES ON
MULTIPLE MEANS
OF ACTION AND
EXPRESSION:
To scaffold the conversion of 24 tenths
to 240 hundredths for students
working below grade level, offer a few
more steps. After verifying that
2 tenths equals 20 hundredths, ask,
“5 tenths is equivalent to how many
hundredths? (50.) 10 tenths is
equivalent to how many hundredths?
(100.) 20 tenths is equivalent to how
many hundredths? (200.) So,
24 tenths equals…?”
T: (Write 240
100.) Write the equivalent decimal.
S: 2.40 or 2.4.
Repeat with 4.3.
Problem 3: Decompose mixed numbers to express as smaller units.
T: (Write 3.6.) Say this decimal.
S: 3 and 6 tenths.
T: How many tenths are in 3 ones?
S: 30 tenths.
T: How many tenths are in 3.6?
S: 36 tenths.
T: In fraction form and unit form, write how many tenths are equal to 3.6.
S: 3.6 = 36 tenths = 36
10.
T: How many hundredths are in 3 ones?
S: 300 hundredths.
T: How many hundredths are in 6 tenths?
S: 60 hundredths.
T: How many hundredths are in 3.6?
S: 360 hundredths.
T: In fraction form and unit form, write how many hundredths are equal to 3.6.
S: 3.6 = 360 hundredths = 360
100.
Repeat this process with 5.2 and 12.5.
Problem Set (10 minutes)
Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not specify a method for solving. Students should solve these problems using the RDW approach used for Application Problems.
MP.6
Lesson 8 NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
118
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Student Debrief (10 minutes)
Lesson Objective: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
The Student Debrief is intended to invite reflection and active processing of the total lesson experience.
Invite students to review their solutions for the Problem Set. They should check work by comparing answers with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the lesson.
Any combination of the questions below may be used to lead the discussion.
Explain why the area model in Problem 1 is a good tool for representing the decimal fraction. How does it help to determine the equivalent decimal number?
How did drawing the place value disks in Problem 2 help you to understand decomposing from one unit to another?
How did solving Problem 3 help you to solve Problem 4?
What strategies did you use when completing the chart in Problem 5? Did you complete one column at a time or one row at a time? Which columns were especially helpful in completing other columns?
How is decomposing hundreds to tens or tens to ones similar to decomposing ones to tenths or tenths to hundredths?
When decomposing numbers on the place value chart, each column to the right of another shows 10 times as many parts. Explain why this is so. Even though we have 10 times as many parts, we are really dividing. Explain.
How did the Application Problem connect to today’s lesson?
Exit Ticket (3 minutes)
After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help with assessing students’ understanding of the concepts that were presented in today’s lesson and planning more effectively for future lessons. The questions may be read aloud to the students.
Lesson 8 Sprint NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
119
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Write Fractions and Decimals
1. 3
10 = . 23. 2 +
1
10 +
6
100 = .
2. 3
100 = . 24. 2 + 0.1 + 0.06 = .
3. 23
100 = . 25. 3 + 0.1 + 0.06 = .
4. 123
100 = . 26. 3 + 0.1 + 0.04 = .
5. 423
100 = . 27. 3 + 0.5 + 0.04 = .
6. 0.07 = 28. 2 + 0.3 + 0.08 = .
7. 1.07 = 29. 2 + 0.08 = .
8. 0.7 = 30. 1 + 0.3 = .
9. 1.7 = 31. 10 + 0.3 = .
10. 1.74 = 32. 1 + 0.4 + 0.06 = .
11. 4
100 = . 33. 10 + 0.4 + 0.06 = .
12. 0.6 = 34. 30 + 0.7 + 0.02 = .
13. 7
100 = . 35. 2 +
3
10 + 0.05 = .
14. 0.02 = 36. 4 + 0.5 + 3
100 = .
15. 9
100 = . 37. 4 +
3
100 + 0.5 = .
16. 10
100 = . 38. 0.5 +
3
100 + 4 = .
17. 10
100 +
2
100 = . 39. 20 + 0.8 + 0.01 = .
18. 1
10 +
2
100 = . 40. 4 +
9
100 +
2
10 = .
19. 1
10 +
3
100 = . 41. 0.04 + 2 + 0.7 =
20. 1
10 +
4
100 = . 42.
6
10 + 8 +
2
100 = .
21. 1
10 +
9
100 = . 43.
5
100 + 8 + 0.9 =
22. 3 + 1
10 +
9
100 = . 44. 0.9 + 10 +
4
100 = .
A Number Correct: _______
Lesson 8 Sprint NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
120
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Write Fractions and Decimals
1. 1
10 = . 23. 2 +
1
10 +
4
100 = .
2. 2
10 = . 24. 2 + 0.1 + 0.04 = .
3. 3
10 = . 25. 3 + 0.1 + 0.04 = .
4. 7
10 = . 26. 3 + 0.1 + 0.06 = .
5. 5
10 = . 27. 3 + 0.5 + 0.06 = .
6. 0.2 = 28. 2 + 0.4 + 0.09 = .
7. 0.3 = 29. 2 + 0.06 = .
8. 0.4 = 30. 1 + 0.5 = .
9. 0.8 = 31. 10 + 0.5 = .
10. 0.6 = 32. 1 + 0.2 + 0.04 = .
11. 4
10 = . 33. 10 + 0.2 + 0.04 = .
12. 0.9 = 34. 30 + 0.9 + 0.06 = .
13. 6
10 = . 35. 2 +
5
10 + 0.07 = .
14. 0.5 = 36. 4 + 0.7 + 5
100 = .
15. 9
10 = . 37. 4 +
5
100 + 0.7 = .
16. 10
10 = . 38. 0.7 +
5
100 + 4 = .
17. 11
10 = . 39. 20 + 0.6 + 0.01 = .
18. 12
10 = . 40. 6 +
7
100 +
4
10 = .
19. 17
10 = . 41. 0.06 + 2 + 0.9 =
20. 27
10 = . 42.
8
10 + 6 +
4
100 = .
21. 47
10 = . 43.
3
100 + 8 + 0.7 =
22. 34
10 = . 44. 0.7 + 10 +
6
100 = .
B
[KEY]
Number Correct: _______
Improvement: _______
Lesson 8 Problem Set NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
121
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Name Date
1. Use the area model to represent 250
100. Complete the number sentence.
a. 250
100 = ______ tenths = ______ ones ______ tenths = __.____
b. In the space below, explain how you determined your answer to part (a).
2. Draw place value disks to represent the following decompositions:
2 ones = ________ tenths 2 tenths = ________ hundredths
1 one 3 tenths = ____ tenths 2 tenths 3 hundredths = ____ hundredths
ones . tenths hundredths
ones . tenths hundredths
ones . tenths hundredths
ones . tenths hundredths
Lesson 8 Problem Set NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
122
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3. Decompose the units to represent each number as tenths.
a. 1 = ____ tenths b. 2 = ____ tenths
c. 1.7 = _____ tenths d. 2.9 = _____ tenths
e. 10.7 = _____ tenths f. 20.9 = _____ tenths
4. Decompose the units to represent each number as hundredths.
a. 1 = _____ hundredths b. 2 = _____ hundredths
c. 1.7 = _____ hundredths d. 2.9 = _____ hundredths
e. 10.7 = _____ hundredths f. 20.9 = _____ hundredths
5. Complete the chart. The first one has been done for you.
Decimal Mixed Number Tenths Hundredths
2.1 21
10
21 tenths
21
10
210 hundredths 210
100
4.2
8.4
10.2
75.5
Lesson 8 Exit Ticket NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
123
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Name Date
1. a. Draw place value disks to represent the following decomposition:
3 ones 2 tenths = ________ tenths
b. 3 ones 2 tenths = ________ hundredths
2. Decompose the units.
a. 2.6 = ____ tenths b. 6.1 = ____ hundredths
ones . tenths hundredths
Lesson 8 Homework NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
124
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 -Great Minds. eureka math.org This file derived from G4-M6-TE-1.3.0-06.2015
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Name Date
1. Use the area model to represent 220
100. Complete the number sentence.
a. 220
100 = ______ tenths = ______ ones ______ tenths = __.____
b. In the space below, explain how you determined your answer to part (a).
2. Draw place value disks to represent the following decompositions:
3 ones = ________ tenths 3 tenths = ________ hundredths
2 ones 3 tenths = ____ tenths 3 tenths 3 hundredths = ____ hundredths
ones . tenths hundredths
ones . tenths hundredths
ones . tenths hundredths
ones . tenths hundredths
Lesson 8 Homework NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
125
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 -Great Minds. eureka math.org This file derived from G4-M6-TE-1.3.0-06.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
3. Decompose the units to represent each number as tenths.
a. 1 = _____ tenths b. 2 = _____ tenths
c. 1.3 = _____ tenths d. 2.6 = _____ tenths
e. 10.3 = _____ tenths f. 20.6 = _____ tenths
4. Decompose the units to represent each number as hundredths.
a. 1 = _____ hundredths b. 2 = _____ hundredths
c. 1.3 = _____ hundredths d. 2.6 = _____ hundredths
e. 10.3 = _____ hundredths f. 20.6 = _____ hundredths
5. Complete the chart. The first one has been done for you.
Decimal Mixed Number Tenths Hundredths
4.1 4 1
10
41 tenths
41
10
410 hundredths 410
100
5.3
9.7
10.9
68.5
Lesson 8 Template NYS COMMON CORE MATHEMATICS CURRICULUM 4•6
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units.
126
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 -Great Minds. eureka math.org This file derived from G4-M6-TE-1.3.0-06.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
area model and place value chart